Question

From the sum of \[3{x^3} + 7{x^2} - 5x + 1\] and \[5{x^3} - 9{x^2} + 7x - 2\] subtract \[ - 6{x^3} + 2{x^2} - 6x - 8\]

Answer 1

Hint: Here we add the two given polynomials by adding the coefficients of the respective terms in the polynomials. Subtract the third polynomial from the sum obtained by subtracting the coefficients of the respective terms.

Complete step-by-step answer:
We are given three polynomials
\[3{x^3} + 7{x^2} - 5x + 1\]..........… (1)
\[5{x^3} - 9{x^2} + 7x - 2\]...........… (2)
\[ - 6{x^3} + 2{x^2} - 6x - 8\]...........… (3)
Firstly, take sum of equation (1) and equation (2)
Adding the two polynomials in equation (1) and (2) we get
\[ \Rightarrow \left( {3{x^3} + 7{x^2} - 5x + 1} \right) + \left( {5{x^3} - 9{x^2} + 7x - 2} \right)\]
Combine the terms with same variables
\[ \Rightarrow \left( {3{x^3} + 5{x^3}} \right) + \left( {7{x^2} - 9{x^2}} \right) + \left( { - 5x + 7x} \right) + \left( { - 2 + 1} \right)\]
Take the variable common from the bracket and add the values in the bracket.
\[ \Rightarrow \left( {3 + 5} \right){x^3} + \left( {7 - 9} \right){x^2} + \left( { - 5 + 7} \right)x + \left( { - 2 + 1} \right)\]
\[ \Rightarrow 8{x^3} + ( - 2){x^2} + 2x + ( - 1)\]
Multiply the signs of plus and minus to write minus
\[ \Rightarrow 8{x^3} - 2{x^2} + 2x - 1\]............… (4)
Equation (4) is the added polynomial obtained by adding equations (1) and (2)
Secondly, subtract equation (3) from equation (4)
Subtracting polynomial from equation (3) from polynomial in equation (4) we get
\[ \Rightarrow \left( {8{x^3} - 2{x^2} + 2x - 1} \right) - \left( { - 6{x^3} + 2{x^2} - 6x - 8} \right)\]
Combine the terms with same variables
\[ \Rightarrow \left( {8{x^3} - ( - 6{x^3})} \right) + \left( { - 2{x^2} - 2{x^2}} \right) + \left( {2x - ( - 6x)} \right) + \left( { - 1 - ( - 8)} \right)\]
We write the product of two negative signs as positive sign
\[ \Rightarrow \left( {8{x^3} + 6{x^3}} \right) + \left( { - 2{x^2} - 2{x^2}} \right) + \left( {2x + 6x} \right) + \left( { - 1 + 8} \right)\]
Take the variable common from the bracket and solve the values in the bracket.
\[ \Rightarrow \left( {8 + 6} \right){x^3} + \left( { - 2 - 2} \right){x^2} + \left( {2 + 6} \right)x + \left( { - 1 + 8} \right)\]
\[ \Rightarrow 14{x^3} + ( - 4{x^2}) + 8x + 7\]
Multiply the signs of plus and minus to write minus
\[ \Rightarrow 14{x^3} - 4{x^2} + 8x + 7\]
Therefore the final polynomial we get after subtracting \[ - 6{x^3} + 2{x^2} - 6x - 8\] from the sum of \[3{x^3} + 7{x^2} - 5x + 1\] and \[5{x^3} - 9{x^2} + 7x - 2\] is\[14{x^3} - 4{x^2} + 8x + 7\].

Note: * A polynomial is an expression with variables and coefficients. General form of a polynomial is given by \[p(x) = a{x^n} + b{x^{n - 1}} + c{x^{n - 2}}.....\] where a, b, c … are the coefficients of the variables \[{x^n},{x^{n - 1}},{x^{n - 2}}\]…